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Variance asymptotics and central limit theorems for volumes of unions of random closed sets

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber
Tomasz Schreiber*
Affiliation:
Nicholas Copernicus University, Toruń
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Ul Chopina 12/18, 87-100 Toruń, Poland. Email address: [email protected]
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Abstract

Let X, X1, X2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X1 ∪ ∙ ∙ ∙ ∪ Xn)) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝd with centres distributed according to a spherically-symmetric heavy-tailed law.

Keywords

Random setscentral limit theoremvariance asymptoticsconvex hullmean width

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 520 - 539
Copyright
Copyright © Applied Probability Trust 2002 

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